# University of Chicago - Geometric Analysis Seminar

#### Department of Mathematics

We welcome all those who are interested to join us on Tuesdays 3:00-4:00 PM. Most of our talks will be in person, but there could be a few on Zoom. To receive the zoom information please write an email to Ao, Ben, Daniel or Yangyang.

### Autumn 2022

#### September 27, 2022 Tue, 3:00-4:00PM, Zoom

Yang Li, MIT

The Thomas-Yau conjecture is concerned with relating the existence question of special Lagrangians to stability conditions and Fukaya categories. After a very brief recap on Floer theory, we will sketch a variational framework to tackle the conjecture, and the main bulk of the talk is on the analytical aspects. It turns out that a very weak minimizer exists under a suitable Thomas-Yau semistability condition, and the main open problem is to prove the Euler Lagrange equation, which is the special Lagrangian condition.

#### October 4, 2022 Tue, 3:00-4:00PM, Eckhart 308

Fedor Manin, UCSB

I will focus on two closely related questions first considered by Gromov in Metric Structures: given a closed manifold M, (1) What is the maximal degree of an L-Lipschitz map M -> M? (2) Can M be efficiently wrapped with "Euclidean wrapping paper"? For example, let M_k be the connected sum of k copies of CP^2. If k ≤ 3, then M_k can be wrapped efficiently, and the maximal degree of an L-Lipschitz map is ~L^4. If k ≥ 4, then M_k cannot be wrapped with maximal efficiency, and the maximal degree is something like $L^4 \log(L)^{-\Theta(1)}$; to attain this bound, a map must have components at many different "frequencies" in a sense that I will explain. In general, which class a manifold belongs to depends on its cohomology ring. This is joint work with Sasha Berdnikov and Larry Guth.

#### October 11, 2022 Tue, 3:00-4:00PM, Eckhart 308

David Sher, DePaul

#### October 18, 2022 Tue, 3:00-4:00PM, Eckhart 308

Connor Mooney, UC Irvine

The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications, and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with Y. Yang.

#### October 25, 2022 Tue, 3:00-4:00PM, Zoom

Zhenhua Liu, Princeton

Let dimensions d≥3, and codimensions c≥3 be positive integers. Set the exceptional set E={3,4}. For d∉E,c arbitrary or d∈E,c≤d, we prove that for every d-dimensional integral (or mod 2) homology class [Σ] on a compact (not necessarily orientable) d+c-dimensional smooth manifold M, there exist open sets Ω_{[Σ]} in the space of Riemannian metrics so that all area-minimizing integral (or mod 2, respectively) currents in [Σ] are singular for metrics in Ω_{[Σ]}. This settles a conjecture of White about the generic regularity of area-minimizing currents. The answer is sharp dimension-wise. As a byproduct, we determine the moduli space of area-minimizing currents (integral or mod 2) near any area-minimizing transverse immersion of dimension d≥3 and codimension c≥3 satisfying an angle condition of asymptotically sharp order in d.

#### November 1, 2022 Tue, 3:00-4:00PM, Eckhart 308

Joseph Neeman, UT Austin

The classical isoperimetric inequality says that a Euclidean ball is the minimal-perimeter way to enclose a given volume in R^n. If we want to enclose multiple volumes, it's natural to believe that the minimal-perimeter way looks like a cluster of soap bubbles, with interfaces made up of pieces of spheres meeting in certain prescribed ways. In the 1990s, Sullivan put forward a precise conjecture about the minimal-perimeter way to enclose n + 1 or fewer volumes in R^n. A seminar work of Hutchings, Morgan, Ritore, and Ros in 2000 solved this conjecture for two volumes in R^3; this was later extended to higher dimensions but remained mostly stuck at two volumes. I will discuss a recent work (joint with E. Milman) which makes substantial progress for n or fewer volumes in R^n. In particular, we completely settle the cases of three volumes in R^3 and above, and four volumes in R^4 and above.

#### November 8, 2022 Tue, 3:00-4:00PM, Eckhart 308

Marcos Petrúcio Cavalcante, Universidade Federal de Alagoas

Constant mean curvature surfaces are critical points for the area functional under volume preserving variations. From this variational point of view, it is natural to study the index and its relations to the geometry and topology of these surfaces. In this talk, I will describe some classical and new results in this theme, as well as some open problems.

#### November 15, 2022 Tue, 3:00-4:00PM, Eckhart 308

Antonio De Rosa, UMd

#### November 29, 2022 Tue, 3:00-4:00PM, Eckhart 308

Sven Hirsch, Duke

An interesting feature of General Relativity is the presence of singularities which can happen in even the simplest examples such as the Schwarzschild spacetime. However, in this case the singularity is cloaked behind the event horizon of the black hole which has been conjectured to be generically the case. To analyze this so-called Cosmic Censorship Conjecture Penrose proposed in 1973 a test which involves Hawking's area theorem, the final state conjecture and a geometric inequality on initial data sets (M,g,k). For k=0 this Penrose inequality has been proven by Huisken-Ilmanen and by Bray using different methods, but in general the question is wide open. Huisken-Ilmanen's proof relies on the Hawking mass monotonicity formula under inverse mean curvature flow (IMCF), and the purpose of this talk is to generalize the Hawking mass monotonicity formula to initial data sets. For this purpose, we start with recalling spacetime harmonic functions and their applications which have been introduced together with Demetre Kazaras and Marcus Khuri in the context of the spacetime positive mass theorem.

#### November 29, 2022 Tue, 4:15-5:15PM, Eckhart 308

Let $\sigma _1$ be the first Steklov eigenvalue on an embedded free boundary minimal surface in $\b ^3$. We show that an embedded free boundary minimal surface $\Sigma_{\bf g}$ of genus $1 \leq {\bf g} \in \mathbb{N}$, one boundary component and anti-prismatic symmetry satisfy $\sigma_1 (\Sigma _{\bf g}) =1$. In particular, the family constructed by Kapouleas-Wiygul satisfies a such condition. This is a joint work with J.A. Gálvez and J. Pérez.